Match ends, Brighton and Hove Albion 2, Fulham 2.
Brighton manager Chris Hughton praised the character of his team as they fought back from a two-goal deficit to draw with Fulham.
Glenn Murray struck twice in the second half as the Seagulls secured a point that seemed unlikely with 25 minutes remaining.
"To come from 2-0 down here and to show the character we did speaks volumes," Hughton said.
Having missed a penalty, and trailing to goals from Andre Schurrle and Aleksandar Mitrovic, Murray converted Anthony Knockaert's pass to initiate the fightback.
The forward then dispatched their second penalty of the afternoon with six minutes remaining after Mitrovic was penalised for controlling the ball with his arm in the Fulham area.
Pascal Gross had earlier missed a spot-kick for Brighton, with Fulham keeper Marcus Bettinelli turning his effort away.
"We lost momentum after we missed the first penalty," said Hughton. "When you're 2-0 down you're always grateful for what you can get."
Slavisa Jokanovic's side capitalised on their good fortune by going ahead when Schurrle scored from Jean Michael Seri's exquisite scooped pass.
Mitrovic's fourth goal in three games looked to have sealed all three points for the visitors and their first consecutive Premier League wins since April 2014.
However, Brighton, who spurned several good opportunities, persevered to maintain an unbeaten home league record that now stretches back to January.
The outcome left both clubs in the position they had occupied at kick-off, the Cottagers 11th in the table and Brighton one place lower in 12th.
Form going into this game suggested it would be a contest between evenly matched sides, and so it proved as the hosts levelled the scores with six minutes remaining.
Manchester United (twice), Arsenal and Tottenham have all visited the Amex Stadium since Hughton's side last lost at home in the Premier League - a hefty 4-0 reverse to Chelsea - on 20 January.
That resilience had led to Hughton largely keeping faith with the players that ensured top-flight survival last season; of 10 summer signings only Spanish defender Martin Montoya made the starting XI.
Murray was one of those key performers last season. He proved his worth again here in a performance just about running the fine line between combative and undisciplined as he scored for the second home league game in succession.
His nuisance value unnerved the visiting defenders throughout, winning Brighton's first penalty and almost adding to his tally of two goals on three other occasions.
While his late foul on Seri was unwise when already on a booking, his contribution ultimately proved the difference between defeat and a valuable point.
After opening their campaign with consecutive defeats, Fulham looked to have adjusted to the demands of the Premier League with a 4-2 victory against Burnley in their previous outing.
And that feeling was reinforced when they took a two-goal lead on the south coast, albeit slightly fortuitously.
The Cottagers spent more than £100m strengthening their squad in the summer, bringing 13 new players into a squad that earned promotion from the Championship last season.
And key components of that summer recruitment spree all made an impact against Brighton.
Seri, a £25m addition from Nice, superbly created the opening goal for Schurrle before Mitrovic took centre stage.
His bustling presence drew a mistake from Brighton captain Lewis Dunk, enabling him to register his 16th league goal since his debut for the club in February - no striker has scored more during that period across the top four tiers of English football.
However, his late aberration, controlling the ball with his arm while under little pressure in his penalty area, cost his team victory.
"He didn't need to ask for any apologies. I interpret it as bad luck, a mistake," said Fulham manager Jokanovic.
"I don't need to explain to him that he doesn't need to touch the ball in this situation. He's a strong enough lad to keep going, he made one mistake, OK.
"And from another side, he scored the goal, he was fighting really well, he played at the level like he started in the Premier League and I don't want to complain and point the finger at my player, I cannot find any sense (in doing so)."
Brighton travel across the south coast to face Southampton in their next Premier League encounter on 17 September (20:00 BST).
Meanwhile, Fulham visit Manchester City on 15 September (15:00 BST) on their return to top-flight action.
Match ends, Brighton and Hove Albion 2, Fulham 2.
Second Half ends, Brighton and Hove Albion 2, Fulham 2.
Foul by Calum Chambers (Fulham).
Jürgen Locadia (Brighton and Hove Albion) wins a free kick on the left wing.
Maxime Le Marchand (Fulham) is shown the yellow card for a bad foul.
Foul by Maxime Le Marchand (Fulham).
Alireza Jahanbakhsh (Brighton and Hove Albion) wins a free kick in the defensive half.
Attempt missed. Martín Montoya (Brighton and Hove Albion) right footed shot from the right side of the box misses to the left.
Attempt saved. Glenn Murray (Brighton and Hove Albion) header from the centre of the box is saved in the bottom left corner. Assisted by Martín Montoya with a cross.
Offside, Brighton and Hove Albion. Shane Duffy tries a through ball, but Alireza Jahanbakhsh is caught offside.
Attempt saved. Ryan Sessegnon (Fulham) right footed shot from the centre of the box is saved in the centre of the goal. Assisted by Aleksandar Mitrovic.
Substitution, Fulham. Calum Chambers replaces André Schürrle.
Maxime Le Marchand (Fulham) wins a free kick in the defensive half.
Foul by Alireza Jahanbakhsh (Brighton and Hove Albion).
Foul by Aleksandar Mitrovic (Fulham).
Shane Duffy (Brighton and Hove Albion) wins a free kick in the defensive half.
Goal! Brighton and Hove Albion 2, Fulham 2. Glenn Murray (Brighton and Hove Albion) converts the penalty with a right footed shot to the centre of the goal.
Aleksandar Mitrovic (Fulham) is shown the yellow card.
Penalty conceded by Aleksandar Mitrovic (Fulham) with a hand ball in the penalty area.
Stefan Johansen (Fulham) is shown the yellow card.
Foul by Denis Odoi (Fulham).
Glenn Murray (Brighton and Hove Albion) wins a free kick on the left wing.
Offside, Brighton and Hove Albion. Dale Stephens tries a through ball, but Jürgen Locadia is caught offside.
Jean Michael Seri (Fulham) wins a free kick on the left wing.
Foul by Glenn Murray (Brighton and Hove Albion).
Attempt saved. Jürgen Locadia (Brighton and Hove Albion) header from the centre of the box is saved in the centre of the goal. Assisted by Martín Montoya with a cross.
Attempt missed. Kevin McDonald (Fulham) right footed shot from outside the box is too high. Assisted by Jean Michael Seri.
Substitution, Brighton and Hove Albion. Alireza Jahanbakhsh replaces Anthony Knockaert.
Substitution, Brighton and Hove Albion. Jürgen Locadia replaces Davy Pröpper.
Substitution, Fulham. Ryan Sessegnon replaces Luciano Vietto.
Substitution, Fulham. Stefan Johansen replaces André-Frank Zambo Anguissa.
Offside, Fulham. Maxime Le Marchand tries a through ball, but Aleksandar Mitrovic is caught offside.
Attempt blocked. André Schürrle (Fulham) right footed shot from the left side of the box is blocked. Assisted by Alfie Mawson.
Attempt missed. Yves Bissouma (Brighton and Hove Albion) right footed shot from outside the box is too high.
Goal! Brighton and Hove Albion 1, Fulham 2. Glenn Murray (Brighton and Hove Albion) right footed shot from the centre of the box to the bottom left corner. Assisted by Anthony Knockaert.
Aleksandar Mitrovic (Fulham) wins a free kick in the defensive half.
Foul by Lewis Dunk (Brighton and Hove Albion).
Attempt missed. Glenn Murray (Brighton and Hove Albion) header from the centre of the box misses to the left. Assisted by Anthony Knockaert with a cross following a set piece situation.
Foul by Maxime Le Marchand (Fulham).
Glenn Murray (Brighton and Hove Albion) wins a free kick on the right wing.